Distributed saddle-point subgradient algorithms with Laplacian averaging

TL;DR

This paper introduces distributed subgradient algorithms with Laplacian averaging for solving min-max problems with agreement constraints, ensuring convergence to saddle points under certain network conditions.

Contribution

It proposes a novel distributed primal-dual subgradient method with Laplacian averaging for convex-concave saddle-point problems with agreement constraints, and proves convergence under periodic connectivity.

Findings

Convergence of local estimates to saddle points under periodic network connectivity.

Evaluation error decreases at a rate proportional to 1/√t with proper step-size selection.

Simulation demonstrates effectiveness in nonlinear constrained multi-agent optimization.

Abstract

We present distributed subgradient methods for min-max problems with agreement constraints on a subset of the arguments of both the convex and concave parts. Applications include constrained minimization problems where each constraint is a sum of convex functions in the local variables of the agents. In the latter case, the proposed algorithm reduces to primal-dual updates using local subgradients and Laplacian averaging on local copies of the multipliers associated to the global constraints. For the case of general convex-concave saddle-point problems, our analysis establishes the convergence of the running time-averages of the local estimates to a saddle point under periodic connectivity of the communication digraphs. Specifically, choosing the gradient step-sizes in a suitable way, we show that the evaluation error is proportional to $1/\sqrt{t}$, where $t$ is the iteration step. We illustrate our results in simulation for an optimization scenario with nonlinear constraints coupling the decisions of agents that cannot communicate directly.